Cluster formation in precompound nuclei in the time-dependent framework

Background: Modern applications of nuclear time-dependent density functional theory (TDDFT) are often capable of providing quantitative description of heavy ion reactions. However, the structures of precompound (preequilibrium, prefission) states produced in heavy ion reactions are difficult to assess theoretically in TDDFT as the single-particle density alone is a weak indicator of shell structure and cluster states.

Purpose: We employ the time-dependent nucleon localization function (NLF) to reveal the structure of precompound states in nuclear reactions involving light and medium-mass ions. We primarily focus on spin saturated systems with $N=Z$. Furthermore, we study reactions with oxygen and carbon ions, for which some experimental evidence for $\alpha$ clustering in precompound states exists.

Method: We utilize the symmetry-free TDDFT approach with the Skyrme energy density functional UNEDF1 and compute the time-dependent NLFs to describe ${}^{16}\mathrm{O}$ + ${}^{16}\mathrm{O},{}^{40}\mathrm{Ca}$ + ${}^{16}\mathrm{O},{}^{40}\mathrm{Ca}$ + ${}^{40}\mathrm{Ca}$, and ${}^{16,18}\mathrm{O}$ + ${}^{12}\mathrm{C}$ collisions at energies above the Coulomb barrier.

Results: We show that NLFs reveal a variety of time-dependent modes involving cluster structures. For instance, the ${}^{16}\mathrm{O}$ + ${}^{16}\mathrm{O}$ collision results in a vibrational mode of a quasimolecular $\alpha \text{−}{}^{12}\mathrm{C}\text{−}{}^{12}\mathrm{C}\text{−}\alpha$ state. For heavier ions, a variety of cluster configurations are predicted. For the collision of ${}^{16,18}\mathrm{O}$ + ${}^{12}\mathrm{C}$, we showed that the precompound system has a tendency to form $\alpha$ clusters. This result supports the experimental findings that the presence of cluster structures in the projectile and target nuclei gives rise to strong entrance channel effects and enhanced $\alpha$ emission.

Conclusion: The time-dependent nucleon localization measure is a very good indicator of cluster structures in complex precompound states formed in heavy-ion fusion reactions. The localization reveals the presence of collective vibrations involving cluster structures, which dominate the initial dynamics of the fusing system.

Figure 1

Snapshots of ${}^{16}\mathrm{O}$ + ${}^{16}\mathrm{O}$ (top) and ${}^{16}\mathrm{O}$ + ${}^{40}\mathrm{Ca}$ (bottom) TDDFT collision simulations. Total densities normalized to the nuclear saturation density ${\rho }_0=0.16{\mathrm{fm}}^{-3}$ are shown in the left panels while the corresponding localizations ${\mathcal{C}}_{\alpha }$ are displayed in the right panels. Since the collisions are central, axial symmetry with respect to the $z$ axis is conserved.

Figure 2

Localization ${\mathcal{C}}_{\alpha }$ for the central collision of ${}^{16}\mathrm{O}$ + ${}^{16}\mathrm{O}$ at $E_{\mathrm{cm}}=20\mathrm{MeV}$. The numbers indicate the collision time (in fm/$c)$. The black line marks the $\rho =0.05{\mathrm{fm}}^{-3}$ contour of the total density. See Supplemental Material [49] for animations.

Figure 3

Similar to Fig. 2 except for $E_{\mathrm{cm}}=100\mathrm{MeV}$.

Figure 4

The $\alpha \text{−}{}^{12}\mathrm{C}\text{−}{}^{12}\mathrm{C}\text{−}\alpha$ structure formed at $t=330\mathrm{fm}/c$ in ${}^{16}\mathrm{O}$ + ${}^{16}\mathrm{O}$ collision at $E_{\mathrm{cm}}=20\mathrm{MeV}$ for three values of the impact parameter: $b=0\mathrm{fm}$ (a), $b=2\mathrm{fm}$ (b), and $b=4\mathrm{fm}$ (c). The color scale is 0.55 light red, 0.65 green, 0.75 blue, and 0.85 cyan.

Figure 5

Similar to Fig. 2 except for the central collision of ${}^{40}\mathrm{Ca}$ + ${}^{40}\mathrm{Ca}$ at $E_{\mathrm{cm}}=150\mathrm{MeV}$. See Supplemental Material [49] for animations.

Figure 6

Similar to Fig. 5 except for $E_{\mathrm{cm}}=300\mathrm{MeV}$.

Figure 7

NLF in collision of ${}^{40}\mathrm{Ca}$ + ${}^{40}\mathrm{Ca}$ at $E_{\mathrm{cm}}=300\mathrm{MeV}$ for three values of the impact parameter: $b=0\mathrm{fm}$ at $t=210\mathrm{fm}/c$ (a); $b=3\mathrm{fm}$ at $t=210\mathrm{fm}/c$ (b); and $b=6\mathrm{fm}$ at $t=460\mathrm{fm}/c$ (c).

Figure 8

Similar to Fig. 2 except for the central collision of ${}^{16}\mathrm{O}$ + ${}^{40}\mathrm{Ca}$ at $E_{\mathrm{cm}}=80\mathrm{MeV}$. See Supplemental Material [49] for animations.

Figure 9

NLF ${\mathcal{C}}_{\alpha }$ for the ${}^{16}\mathrm{O}$ + ${}^{40}\mathrm{Ca}$ collision of at $E_{\mathrm{cm}}=200\mathrm{MeV}$ and $b=2\mathrm{fm}$ at different times, as indicated.

Figure 10

Similar to Fig. 2 except for the ${}^{18}\mathrm{O}$ + ${}^{12}\mathrm{C}$ collision at $E_{\mathrm{cm}}=14\mathrm{MeV}$ and $b=2\mathrm{fm}$. See Supplemental Material [49] for animations.